Active and Reactive Power

It might be useful to think about antenna systems, particularly phased arrays
with multiple driven elements, in terms more familiar for power systems, that
is, with active and reactive power. The usual antenna system has a single feed
point, and a single feed line, and some matching components, so the natural
model is to think of impedance matching, transmission line loss, and so forth.
However, many antennas have significant reactive impedance, and the flow of
reactive power through the system can result in surprising losses.

Consider, for example, an antenna with a feedpoint impedance of 50 + 30j ohms,
fed from a transmitter with 50 ohm resistive output impedance and 100 Watts
(this means the generator is putting out 70.7 volts into a 50 ohm load). What
does this really mean as far as the feed line and matching components are concerned?
Ignoring the "impedance transforming" effects of the feed line for
a moment, we've got the following sort of circuit. The load is drawn in the
parallel form because it will relate better to what we're doing later. With
70.7 volts out of the source, the total current through the load will be 1.21
Amps: 1.04 through the resistance, and -0.62j through the reactance (it's inductive,
so the current lags the voltage). Note also that the apparent power (voltage
times current) is only 85.75 watts.. providing a nice example of why you want
source and load impedances matched.

We put in a matching network that "cancels" the 30j impedance of
the antenna with a -30j impedance (a capacitor in this case). (by the way, it's
really a -.00882j admittance (in parallel form) at the antenna and a .00882j
admittance in the network.

An interesting thing is that in the feedline between transmitter (source) and
the matching network, there is just the active power flowing. That power flows
through the next segment of feedline, and is "dissipated" in the load
resistance. The load resistance is a combination of the loss resistance of the
antenna and the radiation resistance (where the power actually winds up being
transmitted over the air).

Between the matching network and the antenna, though, there is also a "reactive"
power flow, as charge/current moves back and forth between the inductance in
the antenna and the matching capacitance in the tuner. No actual power is moving,
because the current and voltage are 90 degrees out of phase. In fact, this current
is the 0.62 Amps we calculated before. All well and good, IF the transmission
line is lossless. Consider, though, what happens if the transmission line has
some loss, represented here by a series resistance. (For most transmission lines
at HF and VHF frequencies, dielectric loss is negligible, and most of the loss
is due to the resistive loss in the conductors, so this is a reasonable model.)

The current for both the active AND the reactive power is flowing through Rloss
B. And, because that current is greater than just the current for the active
power, the system loss is greater. With, say, 1 ohm of Rloss B (corresponding
to less than 0.1 dB loss, in the matched case), we'd be dissipating 1.5 Watts
in RlossB. This, then, is why you want to do your matching at the antenna, not
the transmitter, particularly if the mismatch is quite reactive.